Calculus Examples

$\int_{\frac{π}{4}}^{\frac{5 π}{4}} \sin (x) - \cos (x) d x$

Step 1

Split the single integral into multiple integrals.

$\int_{\frac{π}{4}}^{\frac{5 π}{4}} \sin (x) d x + \int_{\frac{π}{4}}^{\frac{5 π}{4}} - \cos (x) d x$

Step 2

The integral of $\sin (x)$ with respect to $x$ is $- \cos (x)$ .

$- \cos (x)]_{\frac{π}{4}}^{\frac{5 π}{4}} + \int_{\frac{π}{4}}^{\frac{5 π}{4}} - \cos (x) d x$

Step 3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \cos (x)]_{\frac{π}{4}}^{\frac{5 π}{4}} - \int_{\frac{π}{4}}^{\frac{5 π}{4}} \cos (x) d x$

Step 4

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$- \cos (x)]_{\frac{π}{4}}^{\frac{5 π}{4}} - (\sin (x)]_{\frac{π}{4}}^{\frac{5 π}{4}})$

Step 5

Simplify the answer.

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Step 5.1

Substitute and simplify.

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Step 5.1.1

Evaluate $- \cos (x)$ at $\frac{5 π}{4}$ and at $\frac{π}{4}$ .

$(- \cos (\frac{5 π}{4})) + \cos (\frac{π}{4}) - (\sin (x)]_{\frac{π}{4}}^{\frac{5 π}{4}})$

Step 5.1.2

Evaluate $\sin (x)$ at $\frac{5 π}{4}$ and at $\frac{π}{4}$ .

$- \cos (\frac{5 π}{4}) + \cos (\frac{π}{4}) - (\sin (\frac{5 π}{4}) - \sin (\frac{π}{4}))$

Step 5.2

Simplify.

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Step 5.2.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \cos (\frac{5 π}{4}) + \frac{\sqrt{2}}{2} - (\sin (\frac{5 π}{4}) - \sin (\frac{π}{4}))$

Step 5.2.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- \cos (\frac{5 π}{4}) + \frac{\sqrt{2}}{2} - (\sin (\frac{5 π}{4}) - \frac{\sqrt{2}}{2})$

Step 5.3

Simplify.

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Step 5.3.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$- - \cos (\frac{π}{4}) + \frac{\sqrt{2}}{2} - (\sin (\frac{5 π}{4}) - \frac{\sqrt{2}}{2})$

Step 5.3.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$- - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - (\sin (\frac{5 π}{4}) - \frac{\sqrt{2}}{2})$

Step 5.3.3

Multiply $- - \frac{\sqrt{2}}{2}$ .

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Step 5.3.3.1

Multiply $- 1$ by $- 1$ .

$1 \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - (\sin (\frac{5 π}{4}) - \frac{\sqrt{2}}{2})$

Step 5.3.3.2

Multiply $\frac{\sqrt{2}}{2}$ by $1$ .

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - (\sin (\frac{5 π}{4}) - \frac{\sqrt{2}}{2})$

Step 5.3.4

Simplify each term.

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Step 5.3.4.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - (- \sin (\frac{π}{4}) - \frac{\sqrt{2}}{2})$

Step 5.3.4.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - (- \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2})$

Step 5.3.5

Combine the numerators over the common denominator.

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - \frac{- \sqrt{2} - \sqrt{2}}{2}$

Step 5.3.6

Subtract $\sqrt{2}$ from $- \sqrt{2}$ .

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - \frac{- 2 \sqrt{2}}{2}$

Step 5.3.7

Cancel the common factor of $- 2$ and $2$ .

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Step 5.3.7.1

Factor $2$ out of $- 2 \sqrt{2}$ .

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - \frac{2 (- \sqrt{2})}{2}$

Step 5.3.7.2

Cancel the common factors.

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Step 5.3.7.2.1

Factor $2$ out of $2$ .

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - \frac{2 (- \sqrt{2})}{2 (1)}$

Step 5.3.7.2.2

Cancel the common factor.

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - \frac{2 (- \sqrt{2})}{2 \cdot 1}$

Step 5.3.7.2.3

Rewrite the expression.

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - \frac{- \sqrt{2}}{1}$

Step 5.3.7.2.4

Divide $- \sqrt{2}$ by $1$ .

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} - - \sqrt{2}$

Step 5.3.8

Multiply $- - \sqrt{2}$ .

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Step 5.3.8.1

Multiply $- 1$ by $- 1$ .

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + 1 \sqrt{2}$

Step 5.3.8.2

Multiply $\sqrt{2}$ by $1$ .

$\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} + \sqrt{2}$

Step 5.3.9

Combine the numerators over the common denominator.

$\frac{\sqrt{2} + \sqrt{2}}{2} + \sqrt{2}$

Step 5.3.10

Add $\sqrt{2}$ and $\sqrt{2}$ .

$\frac{2 \sqrt{2}}{2} + \sqrt{2}$

Step 5.3.11

To write $\sqrt{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{2 \sqrt{2}}{2} + \sqrt{2} \cdot \frac{2}{2}$

Step 5.3.12

Combine $\sqrt{2}$ and $\frac{2}{2}$ .

$\frac{2 \sqrt{2}}{2} + \frac{\sqrt{2} \cdot 2}{2}$

Step 5.3.13

Combine the numerators over the common denominator.

$\frac{2 \sqrt{2} + \sqrt{2} \cdot 2}{2}$

Step 5.3.14

Reorder the factors of $\sqrt{2} \cdot 2$ .

$\frac{2 \sqrt{2} + 2 \sqrt{2}}{2}$

Step 5.3.15

Add $2 \sqrt{2}$ and $2 \sqrt{2}$ .

$\frac{4 \sqrt{2}}{2}$

Step 6

Cancel the common factor of $4$ and $2$ .

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Step 6.1

Factor $2$ out of $4 \sqrt{2}$ .

$\frac{2 (2 \sqrt{2})}{2}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $2$ out of $2$ .

$\frac{2 (2 \sqrt{2})}{2 (1)}$

Step 6.2.2

Cancel the common factor.

$\frac{2 (2 \sqrt{2})}{2 \cdot 1}$

Step 6.2.3

Rewrite the expression.

$\frac{2 \sqrt{2}}{1}$

Step 6.2.4

Divide $2 \sqrt{2}$ by $1$ .

$2 \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$2 \sqrt{2}$

Decimal Form:

$2.82842712 \dots$